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October  2017, 37(10): 5355-5365. doi: 10.3934/dcds.2017233

Strichartz estimates for $N$-body Schrödinger operators with small potential interactions

Younghun Hong ,

Center for Mathematical Analysis & Computation (CMAC), Yonsei University, Seoul 03722, Korea

* Corresponding author: Younghun Hong

Received  January 2016 Revised  May 2017 Published  June 2017

Fund Project: The author is supported by NRF grant 2015R1A5A1009350

In this paper, we prove Strichartz estimates for $N$-body Schrödinger operators, provided that interaction potentials are small enough. Our tools are new Strichartz estimates with frozen spatial variables, and its improvement in the $V_S^p$-norm of Koch and Tataru [19]. As an application, we prove scattering for $N$-body Schrödinger operators.

Keywords: Many-body Schrödinger equation, Strichartz estimates, scattering.
Mathematics Subject Classification: Primary: 35Q40, 35Q41, 81U10; Secondary: 35B40, 35Q70.
Citation: Younghun Hong. Strichartz estimates for $N$-body Schrödinger operators with small potential interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5355-5365. doi: 10.3934/dcds.2017233
References:
[1]

M. Beceanu and M. Goldberg, Schrödinger dispersive estimates for a scaling-critical class of potentials, Comm. Math. Phys., 314 (2012), 471-481. doi: 10.1007/s00220-012-1435-x. Google Scholar

[2]

J. Bergh and J. Löfström, Interpolation Spaces – An Introduction, Grundlehren der Mathematischen Wissenschaften, 1976, x+207 pp. Google Scholar

[3]

N. Burq, F. Planchon, J. Stalker and S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549. doi: 10.1016/S0022-1236(03)00238-6. Google Scholar

[4]

N. Burq, F. Planchon, J. Stalker and S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004), 1665-1680. doi: 10.1512/iumj.2004.53.2541. Google Scholar

[5]

T. Chen, Y. Hong and N. Pavlović, Global well-posedness of the NLS system for infinitely many fermions, Arch. Ration. Mech. Anal., 224 (2017), 91-123. doi: 10.1007/s00205-016-1068-x. Google Scholar

[6]

T. Chen and N. Pavlović, Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in d=3 based on spacetime norms, Ann. Henri Poincaré, 15 (2014), 543-588. doi: 10.1007/s00023-013-0248-6. Google Scholar

[7]

X. Chen and J. Holmer, On the Klainerman-Machedon conjecture for the quantum BBGKY hierarchy with self-interaction, J. Eur. Math. Soc. (JEMS), 18 (2016), 1161-1200. doi: 10.4171/JEMS/610. Google Scholar

[8]

D. Fujiwara, Remarks on convergence of the Feynman path integrals, Duke Math. J., 47 (1980), 559-600. doi: 10.1215/S0012-7094-80-04734-1. Google Scholar

[9]

M. Goldberg, Dispersive estimates for the three-dimensional Schrödinger equation with rough potentials, Amer. J. Math., 128 (2006), 731-750. doi: 10.1353/ajm.2006.0025. Google Scholar

[10]

M. Goldberg, Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials, Geom. Funct. Anal., 16 (2006), 517-536. doi: 10.1007/s00039-006-0568-5. Google Scholar

[11]

M. Goldberg and W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys., 251 (2004), 157-178. doi: 10.1007/s00220-004-1140-5. Google Scholar

[12]

G. Graf, Asymptotic completeness for $N$-body short-range quantum systems: A new proof, Comm. Math. Phys., 132 (1990), 73-101. doi: 10.1007/BF02278000. Google Scholar

[13]

M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-Ⅱ equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941. doi: 10.1016/j.anihpc.2008.04.002. Google Scholar

[14]

S. Herr, D. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in ${{H}^{\text{1}}}\left( {{\mathbb{T}}^{\text{3}}} \right)$, Duke Math. J., 159 (2011), 329-349. doi: 10.1215/00127094-1415889. Google Scholar

[15]

J.-L. Journe, A. Soffer and C. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604. doi: 10.1002/cpa.3160440504. Google Scholar

[16]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039. Google Scholar

[17]

K. Kirkpatrick, B. Schlein and G. Staffilani, Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, Amer. J. Math., 133 (2011), 91-130. doi: 10.1353/ajm.2011.0004. Google Scholar

[18]

S. Klainerman and M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys., 279 (2008), 169-185. doi: 10.1007/s00220-008-0426-4. Google Scholar

[19]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces Int. Math. Res. Not. , 16 (2007), Art. ID rnm053, 36 pp. doi: 10.1093/imrn/rnm053. Google Scholar

[20]

H. Koch, D. Tataru and M. Vişan, Dispersive equations and nonlinear waves: Generalized Korteweg-de Vries, nonlinear Schrödinger, wave and Schrödinger maps Oberwolfach Seminars, 45 (2014). doi: 10.1007/978-3-0348-0736-4. Google Scholar

[21]

I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math., 155 (2004), 451-513. doi: 10.1007/s00222-003-0325-4. Google Scholar

[22]

W. Schlag, Dispersive estimates for Schrödinger operators: A survey, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., Princeton Univ. Press, 163 (2007), 255–285. Google Scholar

[23]

I. M. Sigal and A. Soffer, The N-particle scattering problem: Asymptotic completeness for short-range systems, Ann. of Math.(3), 126 (1987), 35-108. doi: 10.2307/1971345. Google Scholar

show all references

References:
[1]

M. Beceanu and M. Goldberg, Schrödinger dispersive estimates for a scaling-critical class of potentials, Comm. Math. Phys., 314 (2012), 471-481. doi: 10.1007/s00220-012-1435-x. Google Scholar

[2]

J. Bergh and J. Löfström, Interpolation Spaces – An Introduction, Grundlehren der Mathematischen Wissenschaften, 1976, x+207 pp. Google Scholar

[3]

N. Burq, F. Planchon, J. Stalker and S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549. doi: 10.1016/S0022-1236(03)00238-6. Google Scholar

[4]

N. Burq, F. Planchon, J. Stalker and S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004), 1665-1680. doi: 10.1512/iumj.2004.53.2541. Google Scholar

[5]

T. Chen, Y. Hong and N. Pavlović, Global well-posedness of the NLS system for infinitely many fermions, Arch. Ration. Mech. Anal., 224 (2017), 91-123. doi: 10.1007/s00205-016-1068-x. Google Scholar

[6]

T. Chen and N. Pavlović, Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in d=3 based on spacetime norms, Ann. Henri Poincaré, 15 (2014), 543-588. doi: 10.1007/s00023-013-0248-6. Google Scholar

[7]

X. Chen and J. Holmer, On the Klainerman-Machedon conjecture for the quantum BBGKY hierarchy with self-interaction, J. Eur. Math. Soc. (JEMS), 18 (2016), 1161-1200. doi: 10.4171/JEMS/610. Google Scholar

[8]

D. Fujiwara, Remarks on convergence of the Feynman path integrals, Duke Math. J., 47 (1980), 559-600. doi: 10.1215/S0012-7094-80-04734-1. Google Scholar

[9]

M. Goldberg, Dispersive estimates for the three-dimensional Schrödinger equation with rough potentials, Amer. J. Math., 128 (2006), 731-750. doi: 10.1353/ajm.2006.0025. Google Scholar

[10]

M. Goldberg, Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials, Geom. Funct. Anal., 16 (2006), 517-536. doi: 10.1007/s00039-006-0568-5. Google Scholar

[11]

M. Goldberg and W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys., 251 (2004), 157-178. doi: 10.1007/s00220-004-1140-5. Google Scholar

[12]

G. Graf, Asymptotic completeness for $N$-body short-range quantum systems: A new proof, Comm. Math. Phys., 132 (1990), 73-101. doi: 10.1007/BF02278000. Google Scholar

[13]

M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-Ⅱ equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941. doi: 10.1016/j.anihpc.2008.04.002. Google Scholar

[14]

S. Herr, D. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in ${{H}^{\text{1}}}\left( {{\mathbb{T}}^{\text{3}}} \right)$, Duke Math. J., 159 (2011), 329-349. doi: 10.1215/00127094-1415889. Google Scholar

[15]

J.-L. Journe, A. Soffer and C. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604. doi: 10.1002/cpa.3160440504. Google Scholar

[16]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039. Google Scholar

[17]

K. Kirkpatrick, B. Schlein and G. Staffilani, Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, Amer. J. Math., 133 (2011), 91-130. doi: 10.1353/ajm.2011.0004. Google Scholar

[18]

S. Klainerman and M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys., 279 (2008), 169-185. doi: 10.1007/s00220-008-0426-4. Google Scholar

[19]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces Int. Math. Res. Not. , 16 (2007), Art. ID rnm053, 36 pp. doi: 10.1093/imrn/rnm053. Google Scholar

[20]

H. Koch, D. Tataru and M. Vişan, Dispersive equations and nonlinear waves: Generalized Korteweg-de Vries, nonlinear Schrödinger, wave and Schrödinger maps Oberwolfach Seminars, 45 (2014). doi: 10.1007/978-3-0348-0736-4. Google Scholar

[21]

I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math., 155 (2004), 451-513. doi: 10.1007/s00222-003-0325-4. Google Scholar

[22]

W. Schlag, Dispersive estimates for Schrödinger operators: A survey, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., Princeton Univ. Press, 163 (2007), 255–285. Google Scholar

[23]

I. M. Sigal and A. Soffer, The N-particle scattering problem: Asymptotic completeness for short-range systems, Ann. of Math.(3), 126 (1987), 35-108. doi: 10.2307/1971345. Google Scholar

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